Advanced Topics in Signal Processing Advanced Topics in Signal Processing for Wireless Communications for Wireless Communications
Narayan MandayamNarayan Mandayam
WINLABWINLAB, , Rutgers UniversityRutgers University
www.winlab.rutgers.edu/~narayanwww.winlab.rutgers.edu/~narayan
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IntroductionIntroduction
Wireless Data on the move is the primary driver for innovations in signal processing
Examples of situations include:
Cellular like networks for wireless data (Licensed)
Wireless access to the Internet: Wireless LANs (Unlicensed)
Infostations: Intermittent pockets of high bandwidth on the move (Unlicensed)
Wireless Data Communications characterized by Channel variations (time, frequency, space) due to mobility and propagation effectsMultiple Access Interference from known and unknown entities
Challenges in enabling wireless data communications
Mitigating or Exploiting channel variations
Mitigating Multiaccess interference
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Challenges in Enabling Wireless Data Challenges in Enabling Wireless Data
Exploiting Variations: Opportunistic CommunicationsOpportunities for transmission arise in time, frequency and space
Examples include:
MIMO, Space-Time Coding, Scheduling, Resource Allocation
Signal Processing challenges in opportunistic transmission strategies ?
Knowledge of temporal and spatial variations of wireless channels
Higher carrier frequencies, higher mobility, great no. of unknown parameters
Mitigating Interference: Multiuser Detection Exploit interference structure to design tailored receivers
Examples include:
Cellular 3G, Unlicensed band Wireless LANs
Signal Processing challenges in Multiuser Detection ?
Blind and Adaptive Techniques
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Topics Covered in this TalkTopics Covered in this Talk
Opportunistic CommunicationsPilot Assisted MIMO Channel Estimation
Multiuser Detection Blind Interference Cancellation Techniques for CDMA Systems
Subspace TechniquesSIR Estimation in CDMA Systems
Pilot Assisted Estimation of MIMO Pilot Assisted Estimation of MIMO Fading Channel Response and Fading Channel Response and
Achievable Data RatesAchievable Data Rates
Joint work with Dragan SamardzijaJoint work with Dragan Samardzija
Bell Labs, Lucent TechnologiesBell Labs, Lucent Technologies
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IntroductionIntroduction
Pilot assisted MIMO estimation and its impact on achievable rates
The effects of the estimation error are evaluated for Estimates being available at the receiver only: open loop Estimates are fed back to the transmitter allowing water
pouring optimization: closed loop
Results/Analysis may be interpreted as a study of mismatched receiver and transmitter algorithms in MIMO systems
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System AssumptionsSystem Assumptions
Multiple-input multiple-output (MIMO) wireless systems Frequency-flat time-varying wireless channel with additive white
Gaussian noise (AWGN), i.e., Block fading channel We consider two pilot based approaches for the estimation:
Single pilot symbol per block with variable (from data symbols) power
More than one symbol per block with same (as data symbols) power
Orthogonality between the pilots assigned to different transmit antennas
Maximum-likelihood estimate of the channel response
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Signal ModelSignal Model MIMO communication system that consists of M transmit and N receive
antennas Received spatial vector y
y(k) = H(k) x(k) + n(k) (1)
where y(k) in CN, x(k) in CM, n(k) in CN, H(k) in CN x M
x is transmitted vector, n is AWGN (E [n nH] = No INxN), and H is the MIMO channel response matrix, all corresponding to the time instance k
hnm (k) is the n-th row and m-th column element of the H(k)
corresponds to a SISO channel response between the transmit antenna m and receive antenna n
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Signal Model, contd.Signal Model, contd.
n-th component of the received spatial vectorspatial vector y(k)=[y1(k)…yN(k)]T (i.e., signal at the receive antenna n) is
(2)
gm (k) is the transmitted signal from the m-th transmit antenna, i.e., x(k)=[g1(k) … gM(k)]T .
The channel response H(k) is estimated using a pilot (training) signal that is a part of the transmitted data
Pilot is sent periodically, every K symbol periods
)()()()(1
k n kgkh k y n
M
mmnmn
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Signal Model, contd.Signal Model, contd. At transmitter m, the K-dimensional temporal vectortemporal vector
gm=[gm(1) … gm(K)]T (whose k-th component is gm(k) (in (2))) is
(3)
a dim=A and a p
im=Ap are amplitudes related as Ap= A d d
im is the unit-variance data, and |d pjm|2=1 is the pilot symbol
sdi and sp
im are temporal signaturestemporal signatures, all corresponding to the m-th transmitter;
L is the number of signal dimensions allocated to the pilot, per transmit antenna
Temporal signatures are mutually orthogonal and they could be: canonical waveform - a TDMA-like waveform K-dimensional Walsh sequence - a CDMA-like waveform
PILOT
L
j
pjm
pjm
pjm
DATA
LMK
i
di
dim
dimm
dada
11
ssg
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Signal Model, contd.Signal Model, contd. Rewrite spatial received signal vector as:
y(k) = H(k)(d(k) + p(k)) + n(k) (4)
d(k) =[d1(k) … dM(k)]T is the data bearingdata bearing transmitted
spatial signalspatial signal where
p(k) =[p1(k) … pM(k)]T is the pilot portionpilot portion of the transmitted
spatial signalspatial signal
DATA
LMK
i
di
dim
dimm ksdakd
1
)()(
PILOT
L
j
pjm
pjm
pjmm ksdakp
1
)()(
sd1
dd11
A
sdK-LM
ddK-LM 1
A
datadata
sp11
dp11
Ap
spL1
dpL 1
Ap
pilot pilot +
TX1
sd1
dd1M
A
sdK-LM
ddK-LM M
A
datadata
sp1M
dp1M
Ap
spLM
dpL M
Ap
pilotpilot +
TXM
X M•Pilots are orthogonal between the TxsPilots are orthogonal between the Txs
Data temporal signatures reused across TxsData temporal signatures reused across Txs
MIMO transmitter with M antennas
1g
Mg
ppp
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Model AssumptionsModel Assumptions
Block-fading channel model with channel coherence K Tsym, hnm(k)~hnm,
for k = 1,…, K, for all m and n The elements of H are iid random variables When applying different number of transmit antennas, the total average
transmitted power must be conserved. Per pilot period it is
(5)
Amount of transmitted energy that is allocated to the pilot (percentage wise) is
(6)
L
j
pjm
LMK
i
dimav aa
K
MP
1
2
1
2
[%]1002
2
LLMK
L
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Pilot Arrangements – case 1Pilot Arrangements – case 1 Two different pilot arrangements:
L=1 and Ap= A, single dimension taken by pilot, with different power from data symbols. The data symbol amplitude is
(7)
In SISO systems applied in CDMA wireless systems (e.g., IS-95 and WCDMA)
In MIMO systems, applied in narrowband MIMO implementations [Foschini, Valenzuela, Wolniansky]
Also wideband MIMO implementation based on 3G WCDMA.
M
P
MK
KA av
21
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Pilot Arrangements – case 2Pilot Arrangements – case 2
L > 1 and Ap= A ( = 1), multiple signal dimensions taken by pilot, with the same power as data. The data symbol amplitude is
(8)
Frequently used in SISO systems Wire-line modems Wireless standards (e.g., IS-136 and GSM). Not common practice in MIMO systems.
M
P
MLK
KA av
12
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Estimation of Channel ResponseEstimation of Channel Response Based on previously introduced assumption:
Pilot signatures maintain orthogonality elements of H are iid Background noise is AWGN
Sufficient to estimate hnm (for m=1,…, M, n = 1,…, N) independently
Identical to estimating a SISO channel response between the transmit antenna m and receive antenna n
The estimate of the channel response hnm
L
jn
pjm
pjm
pnm d
LAh
1
H1ˆ rs
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Estimation of Channel Response, contd.Estimation of Channel Response, contd.
(9)
The estimation error is
(10)
corresponds to sample of a white Gaussian random process The channel matrix H estimate is
(11) He is the estimation error matrix Each component of the error matrix He is independent identically
distributed random variable nenm
L
jn
pjm
pjm
pnmnm d
LAhh
1
H1ˆ ns
L
jn
pjm
pjm
p
enm d
LAn
1
H1ns
eHHH ˆ
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Detection and Effective Noise Detection and Effective Noise The sufficient statistics are obtained at the N receive antennas by projecting
the received signal vectors with the corresponding temporal signatures si, i=1,…K-LM
The sufficient statistic for ith signature can be written as
(12)
where E[ni niH] = No INxN
The effective noise vector is
(13)
Covariance matrix of the effective noise vector is
(14)
i
iii
iii AA
dHn
dHdHn
dHHz eeeˆ
i
ii
An dH
ne
H
2
H EE eeH|nHHInnΥΥ
A
NoA ii
i
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Open Loop CapacityOpen Loop Capacity Channel estimates are available to the receiver only Under the assumptions:
Estimate of H has to be stationary and ergodic The channel coding will span across great number of channel blocks Effective noise is treated as independent Gaussian interference
The lower bound for the open loop ergodic capacity is
(15)
(K-LM)/K because L temporal signatures per each transmit antenna allocated to the pilot
1H
x2ˆˆˆ1
detlogE ΥHHIH MK
LMKRC MM
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Comparison to SISO ResultsComparison to SISO Results SISO case [see Shamai, Biglieri, Proakis, IT’98], capacity lower bound
for mismatched decoding as
(16)
where h and are the SISO channel response and its estimate
PropositionFor M = 1, N = 1 (i.e., SISO) the rate R in (17) and R* in (18) are related as
(17)
where is obtained using the pilot assisted estimation Bound in (15) is an extension of the information theoretical bound in (16),
capturing the more specific pilot assisted estimation scheme and generalizing it to the MIMO case
NoPhhE
PhRC
hh
h )|ˆ(|
ˆ1logE
2ˆ|
2
2ˆ*
1x1
h
avPLLK
KPR
K
LKR
2*
)(for
h
Achievable open loop rates vs. power allocated to the pilot, SISO Achievable open loop rates vs. power allocated to the pilot, SISO system, SNR=4, 12, 20dB, K=10, Rayleigh channelsystem, SNR=4, 12, 20dB, K=10, Rayleigh channel
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Capacity Efficiency RatioCapacity Efficiency Ratio
Evaluate performance under optimum pilot power allocation ? For any given SNR, define the capacity efficiency ratio as,
(18)
Maximum rate R is maximized with respect to pilot power Ergodic capacity CMxN, with the ideal knowledge of the
channel response The index M and N correspond to number of transmit and
receive antennas, respectively
NMNM C
R
xx
max
Capacity efficiency ratio vs. channel coherence time length, SISO Capacity efficiency ratio vs. channel coherence time length, SISO system, SNR=4, 12, 20dB, K=10, 20, 40, 100, Rayleigh channelsystem, SNR=4, 12, 20dB, K=10, 20, 40, 100, Rayleigh channel
Pilot arrangement case 1 is more efficient compared to case 2
solid line: channel response estimation
dashed line: ideal channel knowledge
Pilot arrangement case 1
Open loop rates vs. power allocated to the pilot, MIMO system, Open loop rates vs. power allocated to the pilot, MIMO system, SNR=12dB, K=40, Rayleigh channelSNR=12dB, K=40, Rayleigh channel
Capacity efficiency ratio vs. channel coherence time length, Capacity efficiency ratio vs. channel coherence time length, MIMO system, SNR=12dB, K=10, 20, 40, 100, Rayleigh channelMIMO system, SNR=12dB, K=10, 20, 40, 100, Rayleigh channel
Pilot arrangement case 1
1x4 the most efficient
The efficiency is getting smaller as the number of TX antennas grow (for fixed number of received antennas)
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Closed Loop Rates: Mismatched Water PouringClosed Loop Rates: Mismatched Water Pouring H(i-1) and H(i) correspond to the consecutive block faded
channel responses Receiver feeds back the estimate Instead of H(i) , is used to perform the water
pouring transmitter optimization for the i-th block Singular value decomposition (SVD) is performed
For data vector d(k), at the transmitter
(19) S(i) is a diagonal matrix whose elements sjj (j=1, …, M) are
determined by the water pouring algorithm per singular value of
)1(ˆ iH
)1(ˆ)1(ˆ)1(ˆ)1(ˆ H iiii VΣUH
)()()1(ˆ)( kiik dSVd
)1(ˆ iH
)1(ˆ iH
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Mismatched Water Pouring, contd.Mismatched Water Pouring, contd. For diagonal element of (denoted as j = 1, …, M), the
diagonal element of S(i) is defined as
(20)
y0 is a cut-off value that depends on the channel fading statistics
such that the average transmit power stays the same Pav [Goldsmith 93]
Water pouring optimization is on information bearing portion of the signal d(k)
Pilot p(k) is not changed Receiver knows that the transformation in (19) is performed at the
transmitter
)1(ˆ iΣ )1(ˆ ijj
otherwise0
)1(ˆfor)1(ˆ
11
)(0
22
220
2yAi
Aiyisjj
jjjj
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Closed Loop Achievable RatesClosed Loop Achievable Rates Receiver performs estimation resulting in Error matrix Effective noise and its covariance are modified resulting in
(21)
Above application of the water pouring algorithm per eigen mode is suboptimal, i.e., it is mismatched ( is used instead of H(i))
Closed loop system capacity is lower bounded as,
(22)
Assumptions on estimates and effective noise same as before
)()1(ˆ)(ˆˆ iii SVHG
Hˆ2
E eeG|GGGIΥΥ
e
A
NoAWPWP
1H
x2ˆ )(ˆˆ1detlogE WP
MMWPWP
MK
LMKRC ΥGGI
G
)()1(ˆ)( iiiee SVHG
)1(ˆ iH
Ergodic capacity vs. SNR, MIMO system, ideal knowledge of the Ergodic capacity vs. SNR, MIMO system, ideal knowledge of the channel response, Rayleigh channelchannel response, Rayleigh channel
solid line: open loop capacity
dashed: closed loop capacity
Gap between closed loop and open loop is getting smaller for Higher SNR Larger ratio N/M
(number of RX vs. TX antennas)
CDF of capacity, MIMO system, ideal knowledge of the channel CDF of capacity, MIMO system, ideal knowledge of the channel response, Rayleigh channelresponse, Rayleigh channel
solid line: open loop capacity
dashed: closed loop capacity
Closed-loop rates vs. correlation between successive channel Closed-loop rates vs. correlation between successive channel responses, MIMO system, SNR=4dB, K=40, Rayleigh channelresponses, MIMO system, SNR=4dB, K=40, Rayleigh channel
solid line: channel response estimation
dashed: ideal channel response
In both cases delay (temporal mismatch) exists
Pilot arrangement case 1
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Summary of MIMO Pilot EstimationSummary of MIMO Pilot Estimation
Pilot Assisted Channel Estimation for Multiple-input multiple-output wireless systems
Open loop and closed loop ergodic capacity lower bounds are determined
Performance depends on: Percentage of the total power allocated to the pilot Background noise level Channel coherence time length Temporal correlation (for the water pouring)
First pilot-based approach is less sensitive to the fraction of power allocated to the pilot
As the number of transmit antenna increases, the capacity efficiency ratio is lowered (while keeping the same number of receive antennas)